Exact and numerical soliton solutions to nonlinear

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Caspian Journal of Comput ational & Mathematical Engineering 2016 , No.2 (September 2016 ) www.CJComputMathEngin.com

Caspian Journal of Comput ational & Mathematical Engineering ... 2016 , No.2

Cas pian Journal of Computational & Mathematical Engi neering (CJCME)

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Contents

" Exact and numerical soliton soluti ons to nonlinear wave equations ", Kamruzzaman Khan, Henk Koppel aar , M. Ali Ak bar ............................................................................................................................... P.5

" A Survey on Effecti ve Solution of the Boundary Value Problem for Improperly Elli ptic Equations " , S. M .Ali Raeisian P.23


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Exact and numerical soliton solutions to nonlinear wave equations

Kamruzzaman Khan, Depart ment of Mathemat ics, Pabna University of Science & Technology, Pabna6600, Email: [email protected] m Henk Koppel aar, Faculty of Electrical Eng., Mathematics and Co mp. Science, Delft University of Techn., Netherlands. Email: Koppelaar.Hen k@g mail.co m M. Ali Ak bar, Depart ment of Applied Mathematics, University of Rajshahi, Rajshahi-6205, Bangladesh. Email: ali_ [email protected]

AMS 2010 Mathematical Subject Cl assifiaction: 35K99, 35P05, 35P99.

Keywords: Functional variable method; MEE circular rod; Exact solution; Solitary wave; NLEEs.

Abstract: Because most physical systems are inherently nonlinear by nature, mathematical modeling of physical systems often leads to nonlinear evolution equations. Investigating traveling wave solutions of nonlinear evolution equations (NLEEs) plays a significant role in studying such nonlinear physical phenomena. This paper discusses the application of the functional variable method for finding solitary and periodic wave solutions of the longitudinal wave motion NLEE in a nonlinear magneto-electro-elastic (M EE) circular rod. Each of the obtained solutions Caspian Journal of Comput ational & Mathematical Engineering 2016 , No.2 (September 2016 ) www.CJComputMathEngin.com

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contains an explicit function of the variables in the considered equations. The applied method yielded a powerful mathematical tool for solving these nonlinear wave equations without the necessity of a computer algebra system; according the exact hyperbolic solutions are executed numerically, then they show an undesirable unknown numerical conditioning problem. This paper contains a warning against it and information about how to prevent this outcome.

INTRODUCTION Understanding of a wide variety of nonlinear phenomena in different branches of physics and other knowledge domains has been achieved by means of exact integrable model equations, which solving yields a deeper understanding than those models appro ximately their evolution can be described by means of perturbation theory; building upon wave dynamics. But this paper focuses on constructing exact soliton waves and the periodic solutions for nonlinear evolution equations, including their subsequent exploitation for nu merical purposes. This study was originated by Korteweg, together with Ph D candidate, de Vries. Together, they derived, fro m physics first principles, a nonlinear part ial differential equation, known as the Kd V equation. Solutions are best understood by studying waves in shallow waters. The Kd V equation revealed solitary waves (i.e. waves with permanent shape). This seemingly contradicts nature, as a traveling wave commonly dissolves b y radiation, as in the electro magnetic case. The permanency of shape is due to both a balance of dispersion and nonlinear effects. The nonlinear effects derive fro m wave packets, of nearly the same length propagating with their group velocity, while indiv idual component waves in the packet move through the packet with their phase velocity. It can be generally shown that the energy of a wave disturbance is propagated at the group velocity, not at the phase velocity. Long-wave components of a general solution travel faster than the shortwave components, thus the components disperse. The standard linear theory predicts the dispersal of any disturbance other than a purely sinusoidal one. Zabusky and Kruskal (both in 1965 at Bell Labs in Murray Hill, New Jersey, USA) simu lated numerical solutions of the KdV equation [1]. They noticed, using a difference analog for the Kd V equation [2; p. 233], that colliding solitary waves of emerged fro m collisions with all of their original properties intact. They also noted sinusoidal waves evolving into solitary waves with dispersive radiation. Because of this particlesolitons Caspian Journal of Comput ational & Mathematical Engineering 2016 , No.2 (September 2016 ) www.CJComputMathEngin.com

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4]. The name caught on quickly amongst mathematical physicists as the soliton was the f Incidentally, in water tank experiments, conducted in 1834, John Scott Russell observed that solitary waves could pass through each other and subsequently emerge unchanged. As the concept of fundamental particles did not emerge until early in the next century, Russel did conceptualize his findings as such. Nowadays, the phenomenon is viewed fro m a more general point of view; recognizing solitons and their appro ximations in many wave phenomena. The superconducting state is not the sole exception; there are also topological solitons known as vortices and fluxions. On both microscopic and macroscopic levels solitons are discoverable. Taking into account the omniscient inherent nonlinearities in the nature of matter and propagation media, appearance of solitons can be predicted. This paper provides a solution for the nonlinear equation of motion for magnetic excitation and confirms the existence of magnetic solitons. Of course, under strict mathematical defin ition, which implies an infinite life-t ime and an infin ity of conservation laws, exact solitons cannot exist. The mathematical definit ion of solitons, albeit derived from nature by Zabusky and Kruskal, is never fully observable as in nature there is not an infinite lifet ime. The observed 'quasisolitons' are often so long-lived that they are almost 'true solitons'. This is why physicists often generalize the word 'soliton', without full justice to mathematical rigor [5]. Physics, Chemistry, and Biology describe processes and phenomena using theoretical Differential Models endowed with empirical parameters and/or emp irical functions. Exact solutions of those differential equations are preferable for researchers as they enable the design of experiments yielding either a positive or negative correlation via creating appro ximating natural conditions, fro m which estimates are derived as to their parameters and/or functions. This makes the search for exact solutions of NLEEs very significant. The insolvability of model equations, however grounded they empirically are, is potentially a severe drawback. This is where the invention of mathematical methods is critical as it provides exact solutions to the model equations. As a result, many techniques of solving traveling wave equations have been developed over the last three decades, such as, the Tanh Coth function method [6, 7]; the Kudryashov method [8]; the Exp -function method [9 - 12]; the homotopy perturbation method [13 - 15]; the modified simple equation method [16 -expansion method [20 - 24]; the exp-expansion method [25]; the transformed rational function method [26]; Ricatti Ansätze [27]; the mu ltip le exp-function method [28, 29]; the generalized Hirota bilinear method [30]; and the Frobenius integrable decompositions [31]; as well as many others. Each methods has its distinct pluses and minuses. Caspian Journal of Comput ational & Mathematical Engineering 2016 , No.2 (September 2016 ) www.CJComputMathEngin.com

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A unified method applicable to all types of NLEEs would be a scientific breakthrough for all knowledge domains, including Physics. Until then, imp rovement of a particular method to reveal/predict unknown solutions to existing NLEEs, will have to be researched in the various application domains. Zerarka et al. [32] in 2010 proposed a functional variable method to solve NLEEs arising in mathemat ical physics. This pioneering work spawned studies to refine the initial idea [5, 33] This paper applies the functional variable method to construct exact solutions for nonlinear longitudinal wave motion equations in nonlinear magneto -electro-elastic circular rods. We extended the use of the functional variable method by developing a novel solution procedure of simplicity, capable of extension to all NLEEs. The paper is organized as follows: Section 2, the functional variable method; Section 3, application of this method to the nonlinear evolution equation mentioned before; Section 4, results and discussion; and Section 5 Conclusions.

THE ALGORITHM OF THE FUNCTIONAL VARIABLE METHOD Zerarka et al. [32] were the first to propose the functional variable method to solve various NLEEs arising in mathemat ical physics and engineering. First, the NLEE is written in two independent variables x and t, (1) as where the subscript denotes partial derivative, where F is a polynomial in and its partial derivatives and is called a dependent variable or unknown function, to be determined. The main steps of this method are as follows [5, 32-34]:

Step 1. To find the traveling wave solutions of Eq.(1) we introduce the wave variable , so that , where is the wave velocity and k the wave number. This, reduces Eq.(1) into the following ordinary differential equation (ODE): (2) Caspian Journal of Comput ational & Mathematical Engineering 2016 , No.2 (September 2016 ) www.CJComputMathEngin.com

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Where X is a polynomial in

and its derivatives,

u =

du d 2u , u = 2 , and d d

so on. Step 2. We make a transformat ion in which the unknown function considered as a functional variable in the form:

is

(3) and some successively derivatives of

are as follows:

1 2 Y 2 ' 1 1 = Y2 Y = Y2 2 2 u = Y Y' =

u

'

'

Y2

(4) ... ... ... ... ... where

dY '' d 2Y Y = ,Y = 2 du du '

and so on.

Step 3. We substitute Eq.(3) and Eq.(4) into Eq.(2) to reduce it to the fo llo wing ODE:

(5) Step 4: The key idea of Eq.(5) is of special importance because it admits analytical solutions for a large class of nonlinear wave equations. After integration, Eq.(5) provides an expression of Y, and this, in turn with (3), gives appropriate solutions to the original wave equations. In order to illustrate how the method works, we examine some examples in the following section, were previously been treated using other pre-existing methods.


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APPLICATION OF THE FUNCTIONAL VARIABLE METHOD Nowadays in solid mechanics, nonlinear elastic effects on solitary waves are receiving considerable attention. On the basis of classical linear theory, Zhang and Liu [35] solved the nonlinear equations for a thin elastic rod and derived the longitudinal, torsional and flexural waves using Hamilton's variation principle. Liu and Zhang [36] solved the nonlinear wave equation in an elastic rod by the Jacobi elliptic function expansion method. With the increasing usage of magneto-electro-elastic (M EE) structures in various engineering fields, such as actuators, sensors, etc., wave propagation in MEE media has also attracted more researchers. Using the propagator matrix and state -vector approach, Chen et al. [37] presented an analytical treatment for the propagation of harmonic waves in M EE mult ilayered p lates. Based on the constitutive relation for t ransversely isotropic piezoelectric and piezo magnetic materials, comb ined with the differential equations of motion, Xue et al. [38] derived a longitudinal wave motion equation in a Magneto-electro-elastic circular rod of the form,

utt

c02 u z z

c02 2 u + Nut t 2

=0 zz

(6) where c0 is the linear longitudinal wave velocity for a MEE circular rod and N is the dispersion parameter, both depending on the material properties as well as the geometry of the rod. Eq.(6) is a nonlinear wave equation with dispersion caused by the transverse The traveling wave transformation is: , (7)


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Eq.(7) reduces Eq.(6) into the follo wing ODEs:

By integrating Eq.(8) twice with respect to we obtain: 2

u =

c02

Nk

2

2

u

(8) and neglecting the integration constant,

c02 2 Nk 2

2

u2 .

(9) Following Eq.(4), it is easy to deduce from Eq.(9) an expression for the function :

Y2 u =

2

2

c02

Nk 2

2

u

c02 Nk 2

2

u2 .

(10)

Integrating Eq.(10) and setting the constant of integration to zero yields: 2

Y2 u =

Nk

c02 2

2

u2

c02 3Nk 2

2

u3

(11)

or , where

A=

3

2

c02

c02

. (12)

Substituting Eq.(3) into Eq.(12), we obtain


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. (13) Separating the variables in Eq.(13) and then integrating, by setting the constant of integration to zero, y ields:

. (14) Finally, by co mp leting the integration in Eq. (14), two cases of traveling wave solutions of the longitudinal motion equation are obtained (in a nonlinear magnetoelectro-elastic circu lar rod) after a straightforward algebraic manipulat ion. 2 02

If

N

> 0 we obtain the following hyperbolic traveling wave solutions

: 2 1

2 0

2

2 0

2

.

2 0

(15) 2 2

2 0

2 2

2 0

2 0

(16) Eq.(15) and Eq .(16) are solitary wave solutions for the longitudinal wave motion equation of a magneto-electro-elastic circular rod. 2

If

c02 N